A SURVEY OF DISTANCE MAGIC GRAPHS

In this report, we survey results on distance magic graphs and some closely related graphs. A distance magic labeling of a graph G with magic constant k is a bijection l from the vertex set to {1, 2, . . . , n}, such that for every vertex x Σ l(y) = k,y∈NG(x) where NG(x) is the set of vertices of G adjacent to x. If the graph G has a distance magic labeling we say that G is a distance magic graph. In Chapter 1, we explore the background of distance magic graphs by introducing examples of magic squares, magic graphs, and distance magic graphs. In Chapter 2, we begin by examining some basic results on distance magic graphs. We next look at results on different graph structures including regular graphs, multipartite graphs, graph products, join graphs, and splitting graphs. We conclude with other perspectives on distance magic graphs including embedding theorems, the matrix representation of distance magic graphs, lifted magic rectangles, and distance magic constants. In Chapter 3, we study graph labelings that retain the same labels as distance magic labelings, but alter the definition in some other way. These labelings include balanced distance magic labelings, closed distance magic labelings, D-distance magic labelings, and distance antimagic labelings. In Chapter 4, we examine results on neighborhood magic labelings, group distance magic labelings, and group distance antimagic labelings. These graph labelings change the label set, but are otherwise similar to distance magic graphs. In Chapter 5, we examine some applications of distance magic and distance antimagic labeling to the fair scheduling of tournaments. In Chapter 6, we conclude with some open problems.

Construção de quadrados mágicos pelo método do passo uniforme

Lehmer (1929) analisa matematicamente o método do passo uniforme para construção de quadrados mágicos de ordem impar. Ele divide sua análise em várias etapas. Na primeira delas, envolvendo a discussão de condições necessárias e suficientes para o preenchimento do quadrado pelo método, o autor afirma que se dois números guardarem entre si uma certa relação, eles serão designados a ocupar a mesma célula do quadrado causando seu não preenchimento. A análise do preenchimento pelo método do passo uniforme envolve a resolução de um sistema linear módulo n. Nesse trabalho, discutimos o comportamento das soluções desse sistema quando o método falha no preenchimento. Como consequência, concluímos que números que guardam a relação mencionada nunca ocupam a mesma célula. A análise das condições necessárias e suficientes para obter quadrados mágicos segundo a definição de Lehmer (1929) envolve a resolução de equações de congruências lineares a duas variáveis. Nesse trabalho, detalhamos os resultados de Lehmer (1929). A análise das condições necessárias e suficientes para obtenção de quadrados mágicos, como são reconhecidos usualmente, também envolve a resolução de equações de congruências lineares a duas variáveis. Discutimos o comportamento das soluções dessas equações para obter diagonais principais mágicas. Como consequência, mostramos que diagonais principais mágicas são obtidas se e somente se as coordenadas iniciais guardarem certas relações

Exercise sheet 3

Exercises and solutions in LaTex

Exercise sheet 3 pdf

Exercises and solutions in PDF

Mathematical recreations and problems of past and present times.

Some arithmetical questions.--Some geometrical questions.--Some mechanical questions.--Some miscellaneous questions.--Magic squares.--Unicursal problems.--Three geometrical problems.--Astrology.--Hyper-space.--Time and its measurement.--The constitution of matter.

Das Problem der magischen Systeme.

Available on demand as hard copy or computer file from Cornell University Library.

Investigation of different apple cultivars by high resolution magic angle spinning NMR. A feasibility study

(1)H HR-MAS NMR spectroscopy was applied to apple tissue samples deriving from 3 different cultivars. The NMR data were statistically evaluated by analysis of variance (ANOVA), principal component analysis (PCA), and partial least-squares-discriminant analysis (PLS-DA). The intra-apple variability of the compounds was found to be significantly lower than the inter-apple variability within one cultivar. A clear separation of the three different apple cultivars could be obtained by multivariate analysis. Direct comparison of the NMR spectra obtained from apple tissue (with HR-MAS) and juice (with liquid-state HR NMR) showed distinct differences in some metabolites, which are probably due to changes induced by juice preparation. This preliminary study demonstrates the feasibility of (1)H HR-MAS NMR in combination with multivariate analysis as a tool for future chemometric studies applied to intact fruit tissues, e.g. for investigating compositional changes due to physiological disorders, specific growth or storage conditions.

Exploring High-resolution Magic Angle Spinning (HR-MAS) NMR Spectroscopy for Metabonomic Analysis of Apples

Classical liquid-state high-resolution (HR) NMR spectroscopy has proved a powerful tool in the metabonomic analysis of liquid food samples like fruit juices. In this paper the application of (1)H high-resolution magic angle spinning (HR-MAS) NMR spectroscopy to apple tissue is presented probing its potential for metabonomic studies. The (1)H HR-MAS NMR spectra are discussed in terms of the chemical composition of apple tissue and compared to liquid-state NMR spectra of apple juice. Differences indicate that specific metabolic changes are induced by juice preparation. The feasibility of HR-MAS NMR-based multivariate analysis is demonstrated by a study distinguishing three different apple cultivars by principal component analysis (PCA). Preliminary results are shown from subsequent studies comparing three different cultivation methods by means of PCA and partial least squares discriminant analysis (PLS-DA) of the HR-MAS NMR data. The compounds responsible for discriminating organically grown apples are discussed. Finally, an outlook of our ongoing work is given including a longitudinal study on apples.

A second order control-volume finite-element least-squares strategy for simulating diffusion in strongly anisotropic media

An unstructured mesh �nite volume discretisation method for simulating di�usion in anisotropic media in two-dimensional space is discussed. This technique is considered as an extension of the fully implicit hybrid control-volume �nite-element method and it retains the local continuity of the ux at the control volume faces. A least squares function recon- struction technique together with a new ux decomposition strategy is used to obtain an accurate ux approximation at the control volume face, ensuring that the overall accuracy of the spatial discretisation maintains second order. This paper highlights that the new technique coincides with the traditional shape function technique when the correction term is neglected and that it signi�cantly increases the accuracy of the previous linear scheme on coarse meshes when applied to media that exhibit very strong to extreme anisotropy ratios. It is concluded that the method can be used on both regular and irregular meshes, and appears independent of the mesh quality.

Numerical investigations of linear least squares methods for derivative estimation

The results of a numerical investigation into the errors for least squares estimates of function gradients are presented. The underlying algorithm is obtained by constructing a least squares problem using a truncated Taylor expansion. An error bound associated with this method contains in its numerator terms related to the Taylor series remainder, while its denominator contains the smallest singular value of the least squares matrix. Perhaps for this reason the error bounds are often found to be pessimistic by several orders of magnitude. The circumstance under which these poor estimates arise is elucidated and an empirical correction of the theoretical error bounds is conjectured and investigated numerically. This is followed by an indication of how the conjecture is supported by a rigorous argument.

Facile synthesis of Lithium Niobate squares by a combustion route

A combustion synthesis of lithium niobate (LN) squares from activated niobium oxide (Nb2 O5.nH2O) and Li2CO3 was studied to understand all the chemical reactions involved, and the nucleation and square-growth mechanisms. It was found that first the lithium ions react with the fuel (urea), then niobium ions of Nb2 O5.nH2O begin a continuous reaction with the fuel to form metal-organic complexes. LN nuclei are formed by the solid-state reaction of Li- and Nb-organic complexes at 430 degrees celcius. Lithium niobate squares are obtained in the crystallization stasge at 700 degrees celcius, which go on the grow into larger squares at 850 degrees celcius because of the agglomeration effect.